PRESTRESSED CONCRETE STRUCTURES …textofvideo.nptel.ac.in/105106118/lec32.pdf · Department of Civil Engineering, Indian Institute of Technology ... span or as a part of a cantilever

PRESTRESSED CONCRETE STRUCTURES

Amlan K. Sengupta, PhD PE

Department of Civil Engineering,

Indian Institute of Technology Madras

Module – 8: Cantilever and Continuous Beams

Lecture – 32: Cantilever Beams

Welcome back to prestressed concrete structures. This is the first lecture of Module 8 on

cantilever and continuous beams.

(Refer Slide Time: 01:25)

First, we shall have an introduction on cantilever beams. Then we shall move on to the

analysis of cantilevers, the determination of limiting zone and finally, we shall cover

cable profile.


Prestressed cantilever beams are present in buildings and bridges. Usually, the cantilever

is provided with a back span, which is also called anchor span, to reduce the torsion in

the supporting member. In a building, the cantilever can be an extension of a continuous

beam. In a bridge, the cantilever is a part of the “balanced cantilever” girder.


In a building, the cantilever is a part of a beam; like in this figure, the cantilever on the

right has a back span which reduces the moment in the supporting column. Also, the

cantilever can be a part of a continuous beam. Thus, usually we do not find cantilever by

itself, we find it either with a back span or as a part of a cantilever.


A cantilever is a part of a balanced cantilever girder. This is the photograph during the

construction of the Pamban Bridge at Rameshwaram in Tamilnadu. Here you can see that

on each pier there is a balanced cantilever girder, in the sense that there is a cantilever on

each side, so that the eccentric load on the pier is reduced. The cantilever on the two sides

is made in a progressive fashion, so as to have minimal eccentric load on the pier.


The important aspect of the design of prestressed cantilever is the selection of cable

profile. The cable profile refers to the profile of the CGS, which is the variation of the

distance of the CGS from CGC along the length of the beam. When we discussed simply

supported beams, we had talked about cable profile. Once we do the analysis for the

critical section, we find out the eccentricity of the CGS at that section, and then we

determine the cable profile along the length of the beam.

The essential difference between the analysis of a simply supported beam and a

cantilever is in the selection of the cable profile, and for a cantilever it is different than

that of a simply supported beam.


The analysis of a section at a particular location of a prestressed cantilever is similar to

that of a simply supported beam. The difference is that for gravity loads, the bending

moment in cantilever is negative, that is, compression is generated at the bottom of the

beam. Thus, the CGS is placed above the CGC for that particular location.

Whatever we have studied for a simply supported beam, the analysis or the design is

applicable for a cantilever beam as well. But the difference is that for a simply supported

beam the moment is always positive; that means, compression is created at the top due to

bending. For cantilever beams due to gravity loads the moment is negative; that means,

compression is created at the bottom and tension is created at the top. The equations that

we have used for a simply supported beam can be used for a cantilever, provided we take

account of the sign of the moment and then we place the CGS above the CGC. This is the

essential difference in the design of a cantilever beam with respect to that of a simply

supported beam.


The following aspects need to be considered in the analysis and design of a prestressed

cantilever beam. First, certain portions of the back span are subjected to both positive and

negative moments. Hence, there will be two design moments at service loads. In a simply

supported beam, we had seen that throughout the length of the beam, the beam is always

subjected to a positive moment due to gravity loads. But in a back span of a cantilever

beam, the same section can be subjected to a positive moment or a negative moment

depending on the loading condition. Thus, we have two values of design moments for

most of the back span under service loads.


The second aspect is that the beam may be subjected to partial loading and point loading.

A cantilever beam, if it is subjected to a partial loading or a point loading, can generate a

different type of moment condition as compared to when the load is distributed

throughout. Hence, a partial loading or a point loading analysis becomes important for a

cantilever.

The third important aspect is the sequence of loading is important to design the

prestressing force. During construction, the prestressing force can be applied in stages to

take account of the sequence in the loading.

The fourth important aspect is high values of moment and shear occur simultaneously

near the support. For a simply supported beam, the mid span is usually the location of

maximum flexure and a section close to the support is the critical section for shear. But in

a cantilever beam, the section near the support is critical both for shear and moment.

Hence, the section near the support has a larger depth compared to the sections away

from the support. This variation of the depth along the length of the beam should also be

considered in the analysis.


Next, we are studying the important aspects of the analysis of a cantilever beam. The

analysis of a cantilever beam with a back span is illustrated to highlight the aspects stated

earlier. The bending moment diagrams for the following load cases are shown

schematically in the following figures. First, we shall see the bending moment due to the

dead load; next, we shall see the bending moment due to live load only on the back span;

third, we shall see the live load only on the cantilever span and what is the bending

moment due to that; and finally, we shall see the envelop moment diagrams.


This is the sketch of a cantilever on the right side and has back span on the left side; the

dead load is throughout the length of the member. The dead load need not be uniform if

the depth of the member is varying, but for the sake of convenience, right now, we are

showing it as a uniform load. Due to the dead load, the moment in the cantilever is

negative with 0 at the end and with increasing value towards the support. When we go to

the back span then we observe that close to the support we have negative moment, and

then as we move away from the support and proceed toward the left we can have positive

moment in the back span. This is the moment diagram due to the dead load which occurs

throughout the length of the beam.


In the second figure, we are having the live load only on the back span. Due to the live

load, the back span experiences a positive moment. Depending on the variation of the live

load, the location of the maximum moment can be anywhere in the middle of the back

span. Here for convenience we have shown a uniform live load throughout the back span.

Observe that throughout the back span the moment is positive.


Third, the live load is placed only on the cantilever span. Here we find that the cantilever

is subjected to a negative moment which increases from 0 at the end, towards the support

and then in the back span, the moment varies linearly from the support at the right to the

support at the left, where the moment drops down to 0.


If you have the dead load and the live load acting throughout the length of the beam, then

we have a combined moment diagram where the cantilever has a high negative moment.

Then, as we enter the back span, we have negative moment near the support and a

positive moment towards the left support.

Now from these diagrams, suppose, we have the live load on the back span, or the live

load in the cantilever span, or the live load throughout ‒ from these different placements

of the live load, we develop the envelop moment diagrams for the total beam.


In this figure we are seeing the envelop moment diagrams. The orange line is the

maximum moment (Mmax), in an algebraic sense. We see that in the back span we do

have high positive moments towards the left support. Then as we are coming towards the

support on the right, we will not observe any positive moment, but the value of the

negative moment can be less under certain loading conditions. In the cantilever the Mmax

value is always negative depending on the load distribution on the beam.

If we are considering Mmin, which is the minimum value in an algebraic sense, in that

case we find that the moment in the cantilever is substantially high close to the support.

Then, in the back span as we move from right to left, the moment drops down. There may

be a region where there is no negative moment generated which is close to the left

support, but otherwise most of the back span has a negative moment which is given by

the envelop diagram of Mmin. Notice that certain portion of the back span can be

subjected to both a negative moment and a positive moment depending on the loading

conditions.

If I pick up a section somewhere at the middle of the back span, we observe that Mmax is

positive and Mmin is negative. Since the two envelop values have opposite sign, that

means that particular location will experience both positive moment and negative

moment under the service loads.


In the envelop moment diagrams, Mmax and Mmin represent the highest and lowest values,

respectively. These are algebraic values with sign of the moments at a section. Thus,

Mmax is the maximum value and Mmin is the minimum value in an algebraic sense. We are

not comparing the numeric values of the two moments, we are considering the two

moments with the respective signs.

Note that certain portions of the beam are subjected to both positive and negative

moments. This is an important aspect of a cantilever beam with a back span, that there are

locations in the back span which are subjected to positive or negative moments under

service loads.


For moving point loads as in bridges, first the influence line diagram is drawn. The

influence line diagram shows the variation of the moment or shear for a particular

location in the girder, due to the variation of the position of a unit point load.

The concept of influence line diagram is covered in a course on Structural Analysis. In

this lecture we are not covering influence line diagram, but we are mentioning that the

influence line diagram is used if there is a moving point load. Once the influence line

diagram is developed, then the point load is placed to get the worst effect.


Next, we are moving on to the determination of limiting zone, which is done after the

analysis of the beam. The limiting zone of placing the CGS of the tendons is helpful in

selecting a cable profile. Once the structural analysis of a cantilever is performed, the

sectional analysis or design is very similar to the simply supported beam. But the

difference is since the moment has different sign and the sign of the moment can change

based on the location in the beam, selecting an appropriate cable profile is important in

the design of a cantilever beam. To have a proper cable profile, first the limiting zone is

determined and here we shall recollect the concept of limiting zone.

The limiting zone was explained for a simply supported beam under design of members

for flexure. Here, the concept and the equations are first reviewed for a simply supported

beam with positive moment. The same equations can be used for a cantilever beam

provided we are particular about the sign of the moment. Then we can determine the

limiting zone of placing the CGS of the tendons in the cantilever beam.


There are three types of prestressed members as per our code. For a fully prestressed

member which is referred to as Type 1, tension is not allowed under service conditions. If

tension is also not allowed at transfer, the compression in concrete, which will be referred

to as C, always lies within the kern zone. The limiting zone is defined as the zone for

placing the CGS of the tendons such that C always lies within the kern zone. Also the

maximum compressive stresses at transfer and service should be within the allowable

values.

For a simply supported beam under positive moment, we find out the extreme position of

C that is possible without creating any tension for a Type 1 member. The minimum

moment is at transfer and at that time C can be located at the lowest kern point. The

maximum moment is under service loads, when C can be shifted to the upper kern point.

This maximum travel of C is helpful to have an economical section.


For limited prestressed members like Type 2 and Type 3, tension is allowed at transfer

and under service conditions. The limiting zone is defined as the zone for placing the

CGS such that the tensile stresses in the extreme edges are within the allowable values.

Also, the maximum compressive stresses at transfer and service should be within the

allowable values.

The difference of determination of the limiting zone for Type 2 and Type 3 members as

compared to a Type 1 member is that in Type 2 and Type 3 members we allow tensile

stresses at transfer as well as under service loads. Based on the allowable stresses, we

find expressions of the limiting zone. We place the CGS is such a way that C may lie

outside the kern zone provided that the tensile stress in the opposite face is within the

allowable value. Also, the compressive stresses should be within the allowable values for

both at transfer and at service.


The limiting zone is determined from the maximum or minimum eccentricities of the

CGS along the beam corresponding to the extreme positions of C. Thus, once we know

the extreme positions of C, we can determine the corresponding maximum and minimum

eccentricities of the CGS at a particular section. When we draw the loci of the maximum

and the minimum positions along the length of the beam, we determine the limiting zone.

Remember that the limiting zone is related with the CGS of the tendons. Individual

tendons may lie outside the limiting zone; that is, when we are talking of limiting zone,

we are talking about the placement of the CGS within the limiting zone. Individual

tendons may lie outside the limiting zone provided the CGS is lying within the limiting

zone.


For a simply supported beam, the maximum eccentricity, which will be referred to as

emax, at any section corresponds to the lowest possible location of C at transfer. This

generates allowable tensile stress at the top of the section. The maximum compressive

stress at the bottom should also be within the allowable value.


The minimum eccentricity which is represented as emin at any section corresponds to the

highest possible location of C at service, that generates allowable tensile stress at the

bottom of the section. The maximum compressive stress at the top should also be

checked to be within the allowable value.


The values of emax and emin can be determined by equating the stresses at the edges of

concrete within the allowable values. Else, explicit expressions of emax and emin can be

developed. Thus, in order to determine the limiting zone we can calculate emax and emin

for a particular beam, for the given loading conditions, or else we can try to determine

them from some explicit expressions which are discussed here. These expressions help us

to determine the maximum and minimum eccentricities at several locations along the

length of the beam.


The following material gives the expressions of emax and emin for Type 1 and Type 2

sections. The values of emax and emin can be determined at regular intervals along the

length of the beam, from which we shall determine the limiting zone. The zone between

the loci of emax and emin is the limiting zone of the section for placing the CGS.

Here we shall recapitulate the expressions of emax and emin for Type 1 and Type 2

members. For Type 3 members the expressions are similar to the expressions of Type 2

members; the only difference being the value of the allowable tensile stress. Once we

have determined emax and emin at regular intervals along the length of the beam, then we

have the limiting zone, which is in between the loci of emax and emin along the length of

the beam.


We are first referring to a Type 1 section at transfer. The lowest possible location of C is

at the bottom kern point; the tension T is at the level of the CGS. The stress at the top is 0

for a Type 1 section at transfer, and the stress at the bottom is fb. The symbols kt and kb

represent the kern distances or the distances of the kern points from the CGC. The

symbols ct and cb represent the distances of the top and the bottom fibers from the CGC.

Here, emax is the distance of the CGS from the CGC when C lies at the bottom kern point.

We are using the stress diagram to develop the expression of emax.


(emax ‒ kb) is the lever arm by which C has shifted from T. It is equal to the self-weight

moment that is acting as transfer, divided by the prestress at transfer which is denoted as

P0. Transposing kb towards the right side, we have the following expression.

emax = Msw/P0 + kb

Thus, this is the expression of emax for a simply supported beam, for the load condition at

transfer. Using this expression we can calculate the value of emax at a particular location.

We have to check that the stress at the bottom should be less than the allowable

compressive stress at transfer.


Next, we are determining emin for Type 1 section under service loads. We are allowing C

to traverse as high as possible, so that it is at the top kern point under service loads, for

which the stress at the bottom is 0 and the stress at the top is represented as ft. The

location of the CGS for the upper most location of C is emin, which is the minimum

possible eccentricity.


From the stress diagram we can find that the lever arm through which C has shifted is

emin + kt. This is equal to the moment under service loads which is represented as MT

divided by the effective prestress at service which is denoted as Pe. Transposing the term

kt on the right side, we have the following expression.

emin = MT/Pe ‒ kt

Thus, given the values of Mt, Pe and the geometric variable kt we can determine emin at a

particular location. Of course, we need to check that the stress at the top is less than the

allowable compressive stress under service loads. If for a particular section emin comes

out to be a negative, it implies that the CGS can be placed above CGC. This happens near

the supports of a simply supported beam.


Next, we are recollecting the expressions of emax and emin for Type 2 sections. At transfer

there is allowable tensile stress at the top; the position of C is outside the kern region by a

distance e1 and the lever arm between C and T is represented as e2. Here emax is the

position of the CGS, such that C is in the bottom most location which creates allowable

tensile stress at the top during transfer.


The lever arm by which the C traverses is as follows.

emax ‒ kb = (Msw + fct,all Akb)/P0

This expression is a general form of the expression that we have seen for the Type 1

members. Transposing kb on the right side, we have the following expression.

emax = (Msw + fct,all Akb)/P0 + kb

This expression is applicable for both Type 2 and Type 3 members with the appropriate

value of fct,all. Note, that if fct,all is made 0, then this expression becomes same as that for a

Type 1 member. Thus, this expression is more generic as compared to the expression for

a Type 1 member.

Also, we have to check that the stress at the bottom fb is less than the allowable

compressive stress at transfer.


For service conditions, C is at maximum level which can be beyond the kern point. Here

the distance of C from the upper kern point is denoted as e3. The corresponding location

of CGS is emin. The stress at the top is represented as ft and the stress at the bottom is the

allowable tensile stress in the concrete under service conditions. With this stress block,

we can write the expression of emin.

emin + kt = (Mt ‒ fct,all Akt)/Pe

Transposing kt on the right side, we have the following expression.

emin = (Mt ‒ fct,all Akt)/Pe ‒ kt

Thus, this is an explicit expression of emin corresponding to a Type 2 section under

service load condition. The same expression can be used for a Type 3 member if you

have the appropriate value of fct,all. Note, that this expression is a more generic form of

the expression for a Type 1 section, because, when we substitute fct,all = 0, we get the

same expression as for a Type 1 section. Also, we need to check that the stress at the top

should be less than the allowable compressive stress under service condition.

With this recapitulation of the determination of limiting zone for a simply supported

beam, we are moving on to the determination of limiting zone for a cantilever beam.


In a simply supported beam, the external moments are always positive. The minimum

moment is due to self-weight; the maximum moment is under service loads.


For cantilever beams, the minimum external moment need not be at transfer, when the

moment is due to self-weight (MSW); that means, when we are trying to determine the

limiting zone for a cantilever, we have to be careful that the minimum moment may not

be at transfer.

We have to also check the moment conditions under service loads. Under service loads,

there are two moments Mmin and Mmax at a location, which are obtained from the envelop

moment diagrams. Unlike a simply supported beam where we have only one value of the

moment under service condition, in a cantilever beam, we can have two moments Mmax

and Mmin if there are of opposite signs. This is the essential difference between the

analysis of a cantilever beam with respect to that of a simply supported beam.

Thus, we have three moment values at a particular location: one is due to the self-weight

and the other two from the envelop moment diagrams under service loads.


The maximum and minimum eccentricities emax and emin at a particular location are first

determined for service loads from Mmin and Mmax, respectively, at that location and the

effective prestress Pe. Next, we calculate another set of emax and emin for the loads at

transfer, from the self-weight moment MSW and the prestress at transfer P0.


The final emax is the lower of the two values calculated at service and at transfer.

Similarly, the final emin is the higher of the two values calculated at service and at

transfer. Thus, we have two sets of values - one for the service loads and another for the

transfer. The final values are selected judiciously; for emax whichever is lower will satisfy

both the conditions, and hence it is selected as the final emax. Similarly, for emin whichever

is higher will satisfy both the conditions, and we select that as the emin.

The expressions of emax and emin for simply supported beam were developed for positive

moments. For a cantilever beam, corresponding to a negative moment, the eccentricity

implies that the CGS is located above CGC. Thus, when we are using the expressions

from the simply supported beam, we have to be careful that the sign of the moment may

be opposite; in that case, eccentricity refers to the location of CGS above CGC.


To recapitulate the equations, for a Type 1 section at service:

emax = Mmin/Pe + kb

emin = Mmax/Pe ‒ kt


At transfer:

emax = Msw/P0 + kb

emin = Msw/P0 ‒ kt


For a Type 2 section at service:

emax = (Mmin + fct,all Akb)/Pe + kb

emin = (Mmax ‒ fct,all Akt)/Pe + kt


At transfer:

emax = (MSW + fct,all Akb)/P0 + kb

emin = (MSW ‒ fct,all Akt)/P0 ‒ kt

For a Type 3 section, the expressions are same; the only difference is that we have

different values of the allowable tensile stress at transfer and at service.


Next, we are moving on to the cable profiles in a cantilever beam. The cable profiles for a

few beams with cantilever spans are shown schematically in the following figures. The

vertical scale is enlarged to show the location of the CGS with respect to the CGC.


The top one is a beam with a single cantilever. You can observe that since the back span

is subjected to a positive moment, the cable profile is going below the CGC. Near the

support and in the cantilever part, we always have negative moment and hence, the CGS

is above the CGC. Note that the cable profile, which can be selected from the limiting

zone, has a variation close to the moment diagram under uniform load. The difference is

that near the support region, since a tendon cannot have a sharp kink, we are providing an

intermediate curve at the support region without a kink.

If there is a varying cross section in the cantilever span, we can change the location of the

CGS within the limiting zone such that we have minimum bend along the length of the

beam. This helps us in placing the tendon easily, and it reduces the friction losses in a

post-tensioned beam.


For a prismatic beam with uniform cross section along the length, the cable profile is

similar to the moment diagram under uniform load. Thus, for regions of negative moment

the CGS is located above the CGC. Since there cannot be a sharp kink in the tendons and

the supports are not true point supports, the profile is shown curved at the right support.


For a beam with varying depth, the cable profile can be adjusted within emax and emin to

be straight for convenience of layout of the tendons.


For the beam with double cantilever spans and with uniform cross-section, you observe

that the cable profile mimics the moment diagram, where the CGS is below the CGC in

the positive moment region of the middle span. As we go close to the supports, the CGS

is above the CGC in the regions where the moment is negative. In the cantilever spans the

CGS is above the CGC, because the cantilevers are always subjected to negative moment.

If the depth of the beam is varied, then we can select a cable profile within the limiting

zone which is convenient to place and has less loss due to friction. In the second beam

with double cantilevers, a straight tendon can be selected. In fact this straight tendon will

be located above the CGC near the supports, and it will come beneath the CGC near the

middle of the mid span. The selection of a straight cable is more for convenience and to

reduce the friction losses during the post-tensioning operation.


Sometimes we may observe that the top of the beam is varying in elevation. For a beam

with varying level of top of the beam, the cable profile can be selected somewhat like

this, where the CGS is above the CGC in the cantilever portions and in the central region

the CGS is located based on the limiting zone. We are trying to avoid sharp variations in

curvature so as to minimize the losses due to friction.

Thus, the selection of a cable profile is dependent on the limiting zone that we have

determined, and also on the placement of the tendons and friction losses. A cable profile

is selected such that the tendon can be laid conveniently, and there will be minimal

friction loss during the post-tensioning operation.


In today’s lecture, we covered the analysis and design of cantilever beams. First, we

observed the different types of cantilevers that can occur. In a building the cantilever can

be a part of a continuous beam, where the cantilever extends outside the last support. If

there is not a continuous beam, usually a cantilever is provided with a back span to

reduce the torsion in the supporting column. In bridges, there can be cantilevers in a

balanced cantilever type of construction; that means, over a pier two cantilevers are

progressively projecting out during the construction. There can be cantilever spans with

an intermediate span in between. The analysis of a cantilever beam has to be done

carefully by a proper placement of the live load. As compared to a simply supported

beam, the main difference of a cantilever beam is that under different positioning of the

live load, one particular section can have either a positive moment or a negative moment.

If we place the live load in the cantilever span we get a certain moment diagram; if we

place the live load in the back span we get another moment diagram. If we have a live

load through out then it is a third type of moment diagram. From these diagrams, we

determine the envelop moment diagrams, which give a maximum value Mmax and a

minimum value Mmin. These values are algebraic in the sense, that Mmax can have a

positive value or it can have the least negative value, and Mmin can have a least positive

value or the highest negative value.

Once we have done the analysis properly with different positioning of the live load, we

have the design moments Mmax and Mmin for the design of a section of the cantilever.

For the placement of the tendons we use the concept of limiting zone. First we revised the

expressions of the limiting zone for a simply supported beam. We have found that the

maximum eccentricity corresponds to the minimum moment at transfer, and the

minimum eccentricity corresponds to the maximum moment under service loads.

For a cantilever beam, the minimum moment need not be at transfer. We have extra

moment conditions; we have one condition at transfer where the moment is due to the

self-weight, and then under service we have two values of moments Mmax and Mmin for a

particular section which are available from the envelop moment diagrams. Thus, we first

use the Mmax and Mmin values at service to determine one set of emax and emin. Then, we

again calculate emax and emin for transfer, and pick up the values of emax and emin which

satisfy the stress conditions both for transfer and for service. The equations that we have

written are for Type 1 member and Type 2 members. The equations for Type 2 members

are more generic, because if we substitute the allowable tensile stress to be 0, then we get

back the equations for Type 1 member. For Type 3 member the expressions are same as

that for Type 2 member, where we substitute the appropriate value of the allowable

tensile stress.

The expressions for a cantilever are same, but now for service loads we are calculating

emax and emin corresponding to Mmin and Mmax, respectively. Then at transfer we are

calculating another set of emax and emin corresponding to MSW.

Thus, once we have two sets of values for both emax and emin we finally come to the

values of emax and emin which satisfies the stress conditions both at transfer and at service.

Once the limiting zone has been determined, we select the cable profile. The cable profile

can be different type depending on the situation. If we have small beams with uniform

cross section, then the cable profile mimics the moment diagram, where the CGS lies

below the CGC in the positive moment regions, and the CGS lies above CGC in the

negative moment regions. Near the supports the cable profile deviates from the moment

diagram, because we cannot provide a sharp kink and also the supports are not true point

supports.

If the depth of the beam is varying in a large construction, then the profile is adjusted

within the limiting zone such that it is convenient to place the tendons and you have there

is reduced friction loss during the post-tensioning operation. We can try to have a straight

cable profile within the limiting zone, which will be convenient to place the tendon and

will have minimal friction effect. We have seen the cable profiles for a beam with single

cantilever, and also for a beam with two cantilevers on the two sides.

In our next class, we shall move on to the discussion of a continuous beam, which is the

extension of the concept of a cantilever beam, but the number of spans is 2 or 3 or more.

Thank you.

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PRESTRESSED CONCRETE STRUCTURES …textofvideo.nptel.ac.in/105106118/lec32.pdf · Department of Civil Engineering, Indian Institute of Technology ... span or as a part of a cantilever